In a deterministic coupling, \(Z= T(X)\)  for some
deterministic
transformation \(T:\mathbb {R}^n \rightarrow \mathbb {R}^n\)  called a
transport map.
In this case, we say that \(T\)  pushes forward
the prior to the posterior distribution.
For any \(X\sim \pi _{X}\)  and map \(T\) , the pushforward
of \(\pi _{X}\)  by \(T\) 
is given by the law of \(T(X)\)  and is denoted by
\(T_\sharp \pi _{X}\) .
The existence of a map
\(T\)  such that \(T(X) \sim \pi _{X\vert y^*}\) 
is guaranteed under very mild conditions.
For instance, it suffices to have a prior distribution without atoms [1]}.
In fact, there are infinitely many ways to push forward one distribution to another, some of which minimize a notion of integrated cost. Monge's optimal map, for example, minimizes the Euclidean \(L_2\)  cost
\(\mathbb {E}[ \Vert  T(X) - X\Vert ^2 ]\)  over all admissible transformations
[2]}.
Transport maps are interesting because they enable
direct posterior sampling.
That is, if \(x^1,\ldots ,x^M\)  are i.i.d. samples from the prior, then \(T(x^1),\ldots ,T(x^M)\)  are also i.i.d. samples from the posterior
(cf. Figure REF ).
We will see in Section
 that deterministic couplings
are associated with nonlinear generalizations of the square-root EnKF [3]}.
